Cycles, Syllogisms and Semantics: Examining the Idea of Spurious Cycles

The idea that fluctuations can be imparted to a data sequence by passing it through a linear filter has long been familiar to econometricians. It was associated with the discoveries of Slutsky and Yule in the early years of the 20th century. Slutsky (1927, 1937) applied a moving-average filter to random numbers drawn from a public lottery to produce a sequence that had the characteristics of a macroeconomic business cycle. Yule (1927) demonstrated the manner in which a second-order autoregressive model, driven by a white-noise sequence of independently and identically distributed random variables, can give rise to an output that contains cycles of such regularity that one might imagine that they have a mechanical origin. The danger of being misled by an inappropriate use of filters was emphasised by Howrey (1968), who discovered that the long-run economic cycles that Kuznets (1961) claimed to have detected were, in fact, the artefacts of his data processing. It seemed appropriate to describe these cycles as spurious. A linear filter can have two effects. The first effect is to amplify or to attenuate the amplitudes of the cyclical elements of the data to an extent that varies with the frequencies of the elements. This is described as the gain effect of the filter. The second effect is to displace the elements in time, such that their peaks and troughs are advanced or retarded. This is the phase effect of the filter. The phase effect can be avoided if the filter coefficients are disposed symmetrically about a central point, so that the filter reaches equally forward and backwards in time. The examples described so far entail a marked amplification of the amplitudes of sinusoidal elements within a narrow range of frequencies, accompanied by marked attenuations over the remaining frequencies. It is inaccurate to say that the cycles that have been amplified have been induced in the data, since they are already present; but, in these contexts at least, the abuse of language